FANDOM


This blog post contains everything about Dollars Function, including FGH and numbers.

What to solve first?

In order of importance:

• numbers with the lowest & level (Ultimate Array Notation III)

• numbers with least & symbols (Ultimate Array Notation II)

• many &'s solve on lowest to highest & order (Ultimate Array Notation I)

• brackets with the lowest arrow level (Extended Array Notation)

• brackets with the least number of entries (Array Notation)

• brackets with the lowest level (Extended Bracket Notation)

• least nested bracket with the smallest brackets in it (Bracket Notation)

All brackets have the same a$.

Symbols

◆ can be anything

○ is a high-level bracket

◇ is a string of symbols, & and zeroes between them

◈ is a string of symbols

The black square is a group of brackets.

Number Prefixes

These prefixes are needed for groups of numbers.

Grand 2

Great 3

Bigrand 4

Grand Great 5

Bigreat 6

Great Bigrand 7

Grand Bigreat 8

Trigreat 9

Superior 10

Bisuperior 20

Glorious 100

Perfect 1000

Turing Machine

Turing Machine

0 1 _ r 1
1 1 * r *
1 $ 1 r 2
2 _ * r halt
2 1 * r 2
2 [ * r 2
2 ] * l 3
3 * * l 3
3 _ _ r 4
4 1 _ l 5
4 _ _ l 7
4 [ 0 r 7
4 0 0 r 4
5 _ 1 r 6
5 1 1 l 5
6 1 1 r 6
6 _ 1 r 4
7 ] 0 r 9
8 * _ r 8
8 _ * r halt
9 _ * l 10
9 * * * 2
10 * _ l 10
10 1 1 l halt

Defintions of Bracket Notation - Extended Array Notation

Dollar function 1
















FGH Bracket Notation

Bracket Notation FGH Exact value
a$[0] \(f_1(a)\) \(a2\)
a$[1] \(f_2(a)\) \(2^aa\)
a$[2] \(f_3(a)\)
a$[3] \(f_4(a)\)
a$[[0]] \(f_{\omega}(a)\)
a$[0][[0]] = a$a[[0]] \(f_{\omega}(f_1(a))\)
a$[1][[0]] \(f_{\omega}(f_2(a))\)
a$[[0]][[0]] \(f_{\omega}(f_\omega(a))\)
a$[1[0]] \(f_{\omega+1}(a)\)
a$[[0][0]] \(f_{\omega*2}(a)\)
a$[[1]] \(f_{\omega^2}(a)\)
a$[[2]] \(f_{\omega^3}(a)\)
a$[[[0]]] \(f_{\omega^{\omega}}(a)\)
a$[[[[0]]]] \(f_{\omega^{\omega^{\omega}}}(a)\)
a$[[[...[[0]]...]] \(f_{\varepsilon_{0}}(a)\)

Numbers Bracket Notation

Deal Group: 1000$[a]

a means prefix, for example Glorious Deal = 1000$[100]

FGH Extended Bracket Notation

Extended Bracket Notation FGH
\([0]_2\) \(\varepsilon_{0}\)
\([[0]_2]\) \(\varepsilon_{0}+1\)
\([1[0]_2]\) \(\varepsilon_{0}+2\)
\([[0][0]_2]\) \(\varepsilon_{0}+\omega\)
\([[0]_2[0]_2]\) \(\varepsilon_{0}*2\)
\([[[0]_2]]\) \(\omega^{\varepsilon_{0}+1}\)
\([[[[0]_2]]]\) \(\omega^{\omega^{\varepsilon_{0}+1}}\)
\([1]_2\) \(\varepsilon_{1}\)
\([2]_2\) \(\varepsilon_{2}\)
\([[0]]_2\) \(\varepsilon_{\omega}\)
\([[0]_2]_2\) \(\varepsilon_{\varepsilon_{0}}\)
\([0]_3\) \(\zeta_{0}\)
\([[0]_3]\) \(\zeta_{0}+1\)
\([[[0]_3]]\) \(\omega^{\zeta_{0}+1}\)
\([[[[0]_3]]]\) \(\omega^{\omega^{\zeta_{0}+1}}\)
\([[0]_3]_2\) \(\varepsilon_{\zeta_{0}+1}\)
\([[[0]_3]_2]_2\) \(\varepsilon_{\varepsilon_{\zeta_{0}+1}}\)
\([1]_3\) \(\zeta_{1}\)
\([[0]_3]_3\) \(\zeta_{\zeta_{0}}\)
\([0]_4\) \(\eta_{0}\)
\([0]_{[0]}\) \(\varphi(\omega,0)\)
\([0]_{[0]_{[0]}}\) \(\varphi(\varphi(\omega,0),0)\)
\([0,1]\)

\(\varphi(1,0,0)\)

Numbers Extended Bracket Notation

Dollars Group: 1000$[1000]_a


No groups:

BigDollars: 1000$[1000]_[1000]

DollarsBin: 1000$[1000]_[1000]_[1000]

FGH Linear Array Notation

Linear Array Notation FGH
\([0,1]\) \(\varphi(1,0,0)\)
\([1,1]\) \(\varphi(1,0,0)+1\)
\([[0,1],1]\) \(\omega^{\varphi(1,0,0)+1}\)
\([[[0,1],1],1]\) \(\omega^{\omega^{\varphi(1,0,0)+1}}\)
\([[0,1]_2,1]\) \(\varepsilon_{\varphi(1,0,0)+1}\)
\([[0,1]_3,1]\) \(\varphi(2,\varphi(1,0,0)+1)\)
\([[0,1]_{[0,1]},1]\) \(\varphi(\varphi(1,0,0),1)\)
\([0,2]\) \(\varphi(1,0,1)\)
\([0,[0]]\) \(\varphi(1,0,\omega)\)
\([0,[0,1]]\) \(\varphi(1,0,\varphi(1,0,0))\)
\([0,[0,1]_2]\) \(\varphi(1,1,0)\)
\([0,[0,1]_3]\) \(\varphi(1,2,0)\)
\([0,[0,1]_4]\) \(\varphi(1,3,0)\)
\([0,[0,1]_5]\) \(\varphi(1,4,0)\)
\([0,[0,1]_{[0]}]\) \(\varphi(1,\omega,0)\)
\([0,[0,1]_{[0,1]}]\) \(\varphi(1,\varphi(1,0,0),0)\)
\([0,0,1]\) \(\varphi(2,0,0)\)
\([0,1,1]\) \(\varphi(2,0,1)\)
\([0,[0,0,1]_2,1]\) \(\varphi(2,1,0)\)
\([0,0,2]\) \(\varphi(3,0,0)\)
\([0,0,[0,0,1]_2]\)

\(\varphi(1,0,0,0)\)

\([0,0,[0,0,1]_3]\) \(\varphi(1,0,0,0,0)\)
\([0,0,[0,0,1]_{[0]}]\) \(\vartheta(\Omega^\omega)\)
\([0,0,[0,0,1]_{[0,0,1]}]\) \(\vartheta(\Omega^{\varphi(2,0,0)})\)
\([0,0,0,1]\) \(\vartheta(\Omega^\Omega)\)
\([0,1,0,1]\) \(\vartheta(\Omega^\Omega,1)\)
\([0,[0],0,1]\) \(\vartheta(\Omega^\Omega,\omega)\)
\([0,[0,0,0,1]_2,0,1]\) \(\vartheta(\Omega^\Omega+1)\)
\([0,0,1,1]\) \(\vartheta(\Omega^\Omega2)\)
\([0,0,2,1]\) \(\vartheta(\Omega^\Omega3)\)
\([0,0,3,1]\) \(\vartheta(\Omega^\Omega4)\)
\([0,0,[0,0,0,1]_2,1]\) \(\vartheta(\Omega^{\Omega+1})\)
\([0,0,[0,0,0,1]_3,1]\) \(\vartheta(\Omega^{\Omega+2})\)
\([0,0,[0,0,0,1]_{[0,0,0,1]},1]\) \(\vartheta(\Omega^{\Omega+\vartheta(\Omega^\Omega)})\)
\([0,0,0,2]\) \(\vartheta(\Omega^{\Omega2})\)
\([0,0,0,3]\) \(\vartheta(\Omega^{\Omega3})\)
\([0,0,0,[0,0,0,1]_2]\) \(\vartheta(\Omega^{\Omega^2})\)
\([0,0,0,0,1]\) \(\vartheta(\Omega^{\Omega^\Omega})\)
\([0,0,0,0,0,1]\) \(\vartheta(\Omega^{\Omega^{\Omega^\Omega}})\)
\([0 \rightarrow 1]\) \(\vartheta(\varepsilon_{\Omega+1})\)

Bracket Rules

Subscript Bracket Rules

Bracket Rules















FGH Extended Array Notation

Dollar Function FGH
\([0 \rightarrow_2 1]\) \(\vartheta(\varepsilon_{\Omega+1})\)
\([1 \rightarrow_21]\) \(\vartheta(\varepsilon_{\Omega+1})+1\)
\([[0 \rightarrow_2 1] \rightarrow_2 1]\) \(\omega^{\vartheta(\varepsilon_{\Omega+1})+1}\)
\([[0 \rightarrow_21]_2 \rightarrow_2 1]\) \(\varepsilon_{\vartheta(\varepsilon_{\Omega+1})+1}\)
\([0,1 \rightarrow_21]\) \(\vartheta(\Omega,\varepsilon_{\Omega+1})\)
\([0 \rightarrow_2 2]\) \(\vartheta(\varepsilon_{\Omega+1},1)\)
\([0 \rightarrow_2 [0 \rightarrow_21]_2]\)

\(\vartheta(\varepsilon_{\Omega+1}+1)\)

\([0 \rightarrow_2[0 \rightarrow_21]_3]\) \(\vartheta(\varepsilon_{\Omega+1}+2)\)
\([0 \rightarrow_2 0,1]\) \(\vartheta(\varepsilon_{\Omega+1}+\Omega)\)
\([0 \rightarrow_20,2]\) \(\vartheta(\varepsilon_{\Omega+1}+\Omega2)\)
\([0 \rightarrow_2 0,[0 \rightarrow_20,1]_2]\) \(\vartheta(\varepsilon_{\Omega+1}+\Omega^2)\)
\([0 \rightarrow_2 0,0,1]\) \(\vartheta(\varepsilon_{\Omega+1}+\Omega^{\Omega})\)
\([0 \rightarrow_2 0 \rightarrow_2 1]\) \(\vartheta(\varepsilon_{\Omega+1}2)\)
\([0 \rightarrow_20 \rightarrow_2 2]\) \(\vartheta(\varepsilon_{\Omega+1}3)\)
\([0 \rightarrow_20 \rightarrow_2 [0 \rightarrow_2 0 \rightarrow_2 1]_2]\) \(\vartheta(\varepsilon_{\Omega+1}^2)\)
\([0 \rightarrow_{\{2\}} 0 \rightarrow_{\{2\}} 0,1]\) \(\vartheta(\varepsilon_{\Omega+1}^{\Omega})\)
\([0 \rightarrow_20 \rightarrow_2 0,0,1]\) \(\vartheta(\varepsilon_{\Omega+1}^{\Omega^{\Omega}})\)
\([0 \rightarrow_20 \rightarrow_20 \rightarrow_2 1]\) \(\vartheta(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}})\)
\([0 \rightarrow_3 1]\) \(\vartheta(\varepsilon_{\Omega+2})\)
\([0 \rightarrow_{[0]} 1]\) \(\vartheta(\varepsilon_{\Omega+\omega})\)
\([0 \rightarrow_{\{0\}_2} 1]\) \(\vartheta(\varepsilon_{\Omega2})\)
\([0 \rightarrow_{\{0\}_2\{0\}_2} 1]\) \(\vartheta(\varepsilon_{\Omega3})\)
\([0 \rightarrow_{\{\{0\}_2\}} 1]\) \(\vartheta(\varepsilon_{\Omega^2})\)
\([0 \rightarrow_{\{\{\{0\}_2\}\}} 1]\) \(\vartheta(\varepsilon_{\Omega^{\Omega}})\)
\([0 \rightarrow_{\{1\}_2} 1]\) \(\vartheta(\varepsilon_{\varepsilon_{\Omega+1}})\)
\([0 \rightarrow_{\{0\}_3} 1]\) \(\vartheta(\zeta_{\Omega+1})\)
\([0 \rightarrow_{\{0\}_{[0]}} 1]\) \(\vartheta(\varphi(\omega,\Omega+1))\)
\([0 \rightarrow_{\{0,1\}} 1]\) \(\vartheta(\Omega_2)\)
\([0 \rightarrow_{1\{0,1\}} 1]\) \(\vartheta(\varepsilon_{\Omega_2+1})\)
\([0 \rightarrow_{2\{0,1\}} 1]\) \(\vartheta(\varepsilon_{\Omega_2+2})\)
\([0 \rightarrow_{[0]\{0,1\}} 1]\) \(\vartheta(\varepsilon_{\Omega_2+\omega})\)
\([0 \rightarrow_{\{0\}_2\{0,1\}} 1]\) \(\vartheta(\varepsilon_{\Omega_22})\)

\([0 \rightarrow_{\{\{\{0\}_2\}\}\{0,1\}} 1]\)

\(\vartheta(\varepsilon_{\Omega_2^2})\)
\([0 \rightarrow_{\{1\}_2\{0,1\}} 1]\) \(\vartheta(\varepsilon_{\Omega_2^{\varepsilon_{\Omega+1}}})\)
\([0 \rightarrow_{\{0\}_3\{0,1\}} 1]\) \(\vartheta(\varepsilon_{\Omega_2^{\zeta_{\Omega+1}}})\)
\([0 \rightarrow_{\{1,1\}} 1]\) \(\vartheta(\varepsilon_{\Omega_2^{\Omega_2}})\)
\([0 \rightarrow_{\{0,0,1\}} 1]\) \(\vartheta(\varepsilon_{\Omega_2^{\Omega_2^{\Omega_2}}})\)
\([0 \rightarrow_{\{0\rightarrow_{\{2\}} 1\}} 1]\) \(\vartheta(\varepsilon_{\varepsilon_{\Omega_2+1}})\)
\([0 \rightarrow_{\{0\rightarrow_{\{0,1\}}1\}} 1]\) \(\vartheta(\Omega_3)\)
\([0 \rightarrow_{\{0\rightarrow_{\{1,1\}}1\}} 1]\) \(\vartheta(\Omega_4)\)
\([0 \rightarrow_{\{0\rightarrow_{\{[0],1\}}1\}} 1]\) \(\vartheta(\Omega_{\omega})\)
\([0 \rightarrow_{\{0\rightarrow_{\{\{0\}_2,1\}}1\}} 1]\) \(\vartheta(\Omega_{\Omega})\)
\([0 \rightarrow_{\{0\rightarrow_{\{0\rightarrow_{\{0,1\}}1\}}1\}} 1]\) \(\vartheta(\Omega_{\Omega_2})\)
\([0 \rightarrow_{\{0\rightarrow_{\{0\rightarrow_{\{0\rightarrow_{\{0,1\}}1\}}1\}}1\}} 1]\) \(\vartheta(\Omega_{\Omega_{\Omega_2}})\)
\([0 \rightarrow_{\{0\}\text{&}1}1\) \(\psi_I(0)\)

Numbers Linear Array Notation and Extended Array Notation

Gold Group: 1000$[a,a,a...a,a,a] a a's

Gold Rush Group: 1000$[a\(\rightarrow_{a}\)a\(\rightarrow_{a}\)a...a\(\rightarrow_{a}\)a\(\rightarrow_{a}\)a] a a's

Defintions of Ultimate Array Notation

Ultimate Array Notation 1 and 2












FGH Ultimate Array Notation

Dollar Function
\([0 \rightarrow_{\{1\}\text{&}1}1]\) \(\psi_I(1)\)
\([0 \rightarrow_{\{0\}_2\text{&}1}1]\) \(\psi_I(I)\)
\([0 \rightarrow_{\{\{0\}_2\}\text{&}1}1]\) \(\psi_I(I*\omega)\)
\([0 \rightarrow_{\{\{\{0\}_2\}\}\text{&}1}1]\) \(\psi_I(I^2)\)
\([0 \rightarrow_{\{\{\{\{0\}_2\}\}\}\text{&}1}1]\) \(\psi_I(I^I)\)
\([0 \rightarrow_{\{0\}_3\text{&}1}1]\) \(\psi_I(\varepsilon_{I+1})\)
\([0 \rightarrow_{\{0\}_4\text{&}1}1]\) \(\psi_I(\zeta_{I+1})\)
\([0 \rightarrow_{\{0,1\}\text{&}1}1]\) \(\psi_I(\Gamma_{I+1})\)
\([0 \rightarrow_{\{0\}\text{&}2}1]\) \(\psi_{I_2}(0)\)
\([0 \rightarrow_{\{0\}\text{&}3}1]\) \(\psi_{I_3}(0)\)
\([0 \rightarrow_{\{0\}\text{&}\{0\}_2}1]\) \(\psi_{I_I}(0)\)
\([0 \rightarrow_{\{0\}\text{&}_21}1]\) \(\psi_{I(1)}(0)\)
\([0 \rightarrow_{\{0\}\text{&}_31}1]\) \(\psi_{I(2)}(0)\)
\([0 \rightarrow_{\{0\}\text{&}_{\{0\}_2}1}1]\) \(\psi_{I(1,0)}(0)\)
\([0 \rightarrow_{\{0\}\text{&}_{\{0\}_3}1}1]\) \(\psi_{I(2,0)}(0)\)
\([0 \rightarrow_{\{0\}\text{&}_{\{0,1\}}1}1]\) \(\chi(M)\)
\([0 \rightarrow_{\{0\}\text{&}_{\{0\rightarrow_21\}}1}1]\) \(\chi(\varepsilon_{M+1})\)
\([0 \rightarrow_{\{0\}\text{&}_{\{0\rightarrow_{0,1}1\}}1}1]\)

\(\chi(\Gamma_{M+1})\)

\([0 \rightarrow_{\{0\}\text{&}_{\{0\}\text{&}_{\{0,1\}}1}1}1]\) \(\chi(M_2)\)
\([0 \rightarrow_{\{0\}\text{&}_{\{0\}\text{&}_{\{0\}\text{&}_{\{0,1\}}1}1}1}1]\) \(\chi(M_{M_2})\)
treasure function limit of normal Mahlo

Numbers Ultimate Array Notation

Economy Group: 1000$\([0 \rightarrow_{\{0\}\text{&}a} 1]\)

Cash Group: 1000$\([0 \rightarrow_{\{0\}\text{&}_a[0]} 1]\)

Ultimate Array Notation IV and higher

Ultimate Array Notation 3







Ultimate Array Notation 4 and 5










Treasure Function

It marks the ultimate limit of my notation.

Current Definition:

treasure(0) = 1

treasure(a) = a$\([0 \rightarrow_{\{0\}\text{|0|}_{\{0\}\text{|0|}_{\{0\}\text{|0|}_{...}1}1}1} 1]\)

with treasure(a-1) nests

Treasure Numbers

Bronze Treasure = treasure(2)

Silver Treasure = treasure(3)

Emerald Treasure = treasure(4)

Golden Treasure = treasure(5)

Diamond Treasure = treasure(1000)

Legend Treasure = treasure1000(1000)

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